(Redirected from Angular mil)
Not to be confused with Minute of arc.
The PSO-1 reticle in a Dragunov sniper rifle has 10 horizontal lines with 1-mil spacing, which can be used to compensate for wind drift or for range estimation.
Unit system SI derived unit
Unit of Angle
Unit conversions
1 mil in ... ... is equal to ...
turns    1/2000π turn
degrees    ≈ 0.057296°
gons    ≈ 0.063662g
An angle of 1 radian (marked green, approximately 57.3°) corresponds to an angle where the length of the arc (blue) is equal to the radius of the circle (red), while a milliradian therefore corresponds to an angle equal to a thousandth of that value.

A milliradian, often called a mil or mrad (sometimes capitalized MRAD), is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Mils are widely used in adjustment of firearm sights, where an adjustment of 0.1 mil equals 1 cm at 100 meters.

Scopes with mil-dots or marks in the reticle can be used for range estimation if the target size is known (or vice versa to determine a target size if the distance is known). In such applications, the metric units millimeters for target size and meters for target distance is useful, because they coincide with the definition of the milliradian where arc length is defined as 11000 of the radius.

Just like a circle can be divided into 360 degrees or 2π radians, a circle can instead be divided into 2000π ≈ 6283 milliradians. One milliradian approximately equals ≈ 0.057296° or ≈ 3.4377′ (minutes of arc). While the true definition of a mil (2000π ≈ 6283.185… milliradians in a circle) is used in scope adjustment knobs ("turrets") and optical reticles, there are other rounded definitions used for land mapping and artillery which are easier to divide into smaller parts. For instance there are compasses with 6400 NATO mils, 6000 Warsaw Pact mils or 6300 Swedish "streck's" per circle instead of 360°, achieving higher resolution than a 360° compass while also being easier to divide into parts than if true milliradians were used. The term angular mil is used in artillery.[citation needed]

## History

The milliradian (circle/6283.185…) was first used in the mid nineteenth Century by Charles-Marc Dapples (1837–1920), an engineer and professor at the University of Lausanne.[1] Degrees and minutes were the usual units of angular measurement but others were being proposed, with "grads" (circle/400) under various names having considerable popularity in much of northern Europe. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles (60°, circle/6) and hence 600 units to a circle.

Around the time of the start of World War I, France was experimenting with the use of milliemes (circle/6400) for use with artillery sights instead of decigrades (circle/4000). The United Kingdom was also trialing them to replace degrees and minutes. They were adopted by France although decigrades also remained in use throughout World War I. Other nations also used decigrades. The United States, which copied many French artillery practices, adopted mils (circle/6400). Before 2007 the Swedish defence forces used "streck" (circle/6300, streck meaning lines or marks) (together with degrees for some navigation) which is closer to the milliradian but then changed to NATO mils. After the Bolshevik Revolution and the adoption of the metric system of measurement (e.g. artillery replaced "units of base" with meters) the Red Army expanded the 600 unit circle into a 6000 mil one. Hence the Russian mil has nothing to do with milliradians as its origin.

In the 1950s, NATO adopted metric units of measurement for land and general use. Mils, meters, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.

## Mathematical principle

Use of the milliradian is practical because it is concerned with small angles, and at small angles ${\displaystyle \sin \theta \simeq \theta }$. This allows a user to dispense with trigonometry and use simple ratios to determine size and distance with high accuracy for rifle and short distance artillery calculations. More in detail, because ${\displaystyle {\text{subtension}}\simeq {\text{arc length}}}$, instead of finding the angular distance denoted by θ by using the tangent function

${\displaystyle \theta =\arctan {\frac {\text{subtension}}{\text{range}}}}$,

one can instead make a good approximation by using the definition of a radian and the simplified formula:

${\displaystyle \theta ={\frac {\text{subtension}}{\text{range}}}}$

## Firearm sights

The angular mil is commonly used both in the military and civilian shooting sports as a unit for clicks on scope adjustments knobs (turrets), and in optical reticles allowing rough range estimation and precise shot correction. The mils relationship to the trigonometric radian gives rise to the handy property of subtension: One mil approximately subtends one metre at a distance of one thousand metres. More formally the small angle approximation for skinny triangles shows that the angle in radians approximates to the sine of the angle.

Since mil is an angular measurement, the length of the area covered by the angle increase with distance. Mils are very easy to use with metric units. A common scope adjustment increment in European scopes is 0.1 mil, which are sometimes called "centimeter clicks" since 0.1 mil equals 1 cm at 100 meters. Similarly, scopes with 0.2 mil adjustment clicks can be referred to as having "two centimeters clicks", etc.

Angle @ 100 m @ 200 m @ 300 m @ 400 m @ 500 m @ 600 m @ 700 m @ 800 m @ 900 m @ 1000 m
0.1 mil 1 cm 2 cm 3 cm 4 cm 5 cm 6 cm 7 cm 8 cm 9 cm 10 cm
0.2 mil 2 cm 4 cm 6 cm 8 cm 10 cm 12 cm 14 cm 16 cm 18 cm 20 cm
1 mil 10 cm 20 cm 30 cm 40 cm 50 cm 60 cm 70 cm 80 cm 90 cm 100 cm
5 mil 50 cm 100 cm 150 cm 200 cm 250 cm 300 cm 350 cm 400 cm 450 cm 500 cm

Mil adjustment is commonly used in the mechanic adjustment of iron and scope sights in shooting sports, where sight adjustment using mils is particularly useful together with metric units when shooting at regular distances such as 100 m or 300 m, because for instance one click of a sight adjustment of 0.1 mil will move the point of impact exactly 1 cm at 100 m and 3 cm at 300 m respectively. This is not the case when using minutes of arc with imperial units, where one often simplifies 1′ being equal to 1 inch at 100 yards while in reality 1′ at that distance equals 1.047 inches, producing a small error that will increase the more the sight is adjusted or the longer the shooting distance. Therefore, in particular a spotter in long range shooting (i.e. 1000 m and above) theoretically can provide more precise shot corrections using a mil reticle.

### Mil reticles

"FinDot" reticle as used by Finnish Defence Forces snipers (a regular Mil-dot reticle with the addition of 400 m – 1200 m holdover (stadiametric) rangefinding brackets for 1 meter high or 0.5 meter wide targets at 400, 600, 800, 1000 and 1200 m).

Many telescopic sights used on rifles have reticles that are marked in angular mils. This can either be accomplished with lines or dots, and the latter is generally called mil-dots. The mil reticle serves two purposes, range estimation and trajectory correction.

With a mil reticle-equipped scope the distance to an object can be estimated with a fair degree of accuracy by a trained user by determining how many angular mils an object of known size subtends. Once the distance is known, the drop of the bullet at that range (see external ballistics), converted back into angular mils, can be used to adjust the aiming point. Generally mil-reticle scopes have both horizontal and vertical crosshairs marked; the horizontal and vertical marks are used for range estimation and the vertical marks for bullet drop compensation. Trained users, however, can also use the horizontal dots to compensate for bullet drift due to wind. Mil-reticle-equipped scopes are well suited for long shots under uncertain conditions, such as those encountered by military and law enforcement snipers, varmint hunters and other field shooters. These riflemen must be able to aim at varying targets at unknown (sometimes long) distances, so accurate compensation for bullet drop is required.

### Mixing mil and minutes of arc

It is possible to purchase rifle scopes with a mix of for instance a mil reticle and minute-of-arc turrets (or vice versa), but it is general consensus that such mixing should be avoided. It is preferred to either have both a mil reticle and mil adjustment (mil/ mil), or a minute-of-arc reticle and minute-of-arc adjustment to utilize the strength of each system. Then the shooter can know exactly how many clicks to correct based on what he sees in the reticle.

### Mil and minutes of arc conversion table

In the table below conversions from mil to metric values are exact (e.g. 0.1 mil equals exactly 1 cm at 100 meters), while conversions of minutes of arc to both metric and imperial values are approximate.

Minute of arc equivalent
(decimal)
Mil equivalent mm @ 100 m cm @ 100 m in @ 100 m in @ 100 y
1/8′ 0.125′ 0.036 mil 3.64 mm 0.36 cm 0.14 in 0.13 in
0.05 mil 0.172′ 0.05 mil 5 mm 0.5 cm 0.197 in 0.18 in
1/4′ 0.25′ 0.073 mil 7.27 mm 0.73 cm 0.29 in 0.26 in
0.1 mil 0.344′ 0.1 mil 10 mm 1 cm 0.39 in 0.36 in
1/2′ 0.5′ 0.145 mil 14.54 mm 1.45 cm 0.57 in 0.52 in
0.15 mil 0.516′ 0.15 mil 15 mm 1.5 cm 0.59 in 0.54 in
0.2 mil 0.688′ 0.2 mil 20 mm 2 cm 0.79 in 0.72 in
1′ 1.0′ 0.291 mil 29.1 mm 2.91 cm 1.15 in 1.047 in
• 0.1 mil equals exactly 1 cm at 100 m
• 1 mil ≈ 3.44′
• 1′ ≈ 0.291 mil (or 2.91 cm ≈ 3 cm at 100 m)

### Adjustment range and base tilt

The horizontal and vertical adjustment range of a firearm sight is often advertised by the manufacturer using mils. For instance a rifle scope may be advertised as having a vertical adjustment range of 20 mils, which means that by turning the turret the bullet impact can be moved a total of 2 meters at 100 meters (or 4 m at 200 m, 6 m at 300 m etc.). The horizontal and vertical adjustment ranges can be different depending on the particular sight. With a neutral mount, roughly half of the elevation is then usable:

${\displaystyle {\text{usable elevation in neutral mount}}={\frac {\text{scope's total elevation}}{2}}}$

In most regular sport and hunting rifles (except for in long range shooting), rifle scopes are usually mounted without tilt which means that the mounted scope points reasonably parallell to the barrel when it is dialed to the middle of its horizontal adjustment range. This is done because the optical quality of the scope is best in the middle of its adjustment range. While not normally a problem at short and medium range shooting, using a "0 mil" or non-tilted mount means that only about half of the vertical adjustment can be used to compensate for bullet drop. For example, on a scope with a 20 mil vertical elevation range mounted in a level mount, only about 10 mil of the vertical adjustment can be used to compensate for bullet drop at longer ranges.

In long range shooting, tilted scope mounts are often used since it is important to have enough vertical adjustment to compensate for the bullet drop for the given caliber at the given distance. For this purpose scope mounts are sold with varying degrees of tilt, but common values are 3, 6 or 9 mil (10.3′, 20.6′ or 31′ respectitvely) which corresponds to 0.3 m, 0.6 m and 0.9 m at 100 m. If the same 20 mil scope in the example above is mounted with a 9 mil tilt, the scope adjustment has to be bottomed out for short range shooting, but in return the setup will have about 19 mils of vertical adjustment that can be used for bullet drop compensation at long range as opposed to about 10 mils with a neutral mount. Then the maximum scope elevation can be found by:

${\displaystyle {\text{maximum elevation with tilted mount}}={\frac {\text{scope's total elevation}}{2}}+{\text{base tilt}}}$

Total elevation differ between models, but about 10–11 mils are common in hunting scopes, while scopes made for long range shooting usually can have an adjustment range of 30–50 mils.[citation needed] The adjustment range needed to shoot at a certain distance vary with firearm, caliber and load. For example, with a certain .308 load and firearm combination, the bullet may drop 13 mils at 1000 meters (13 meters). To be able to reach out, one could either:

• Use a scope with 26 mils of adjustment in a neutral mount, to get a usable adjustment of 26 mils/2 = 13 mils
• Use a scope with 14 mils of adjustment and a 6 mil tilted mount to achieve a maximum adjustment of 14 mils/2 + 6 = 13 mils

## Range estimation

Estimating mils with hands
Mildot chart as used by snipers

Angle can be used for either calculating target size or range if one of them are known. Where the range is known the angle will give the size, where the size is known then the range is given. When out in the field angle can be measured approximately by using calibrated optics or roughly using one's fingers and hands. With an outstretched arm one finger is approximately 30 mils wide, a fist 150 mils and a spread hand 300 mils.

Mil reticles in optics can easily be used for range estimation because of the precise mathematical simplification that can be made with such small angular measurements, exploiting the attribute of radians that a small angle is a good approximation to its sine, that is, for small angles sin θ ≈ θ. Mil reticles often have dots or marks with a spacing of one mil in between, but graduations can also be finer and coarser (i.e. 0.8 or 1.2 mil).

While a radian is defined as an angle on the unit circle where the arc and radius have equal length, a milliradian is defined as the angle where the arc length is one thousandth of the radius. Therefore, when using milliradians for range estimation, the unit used for target distance needs to be thousand times as large as the unit used for target size. Metric units are particularly useful in conjunction with a mil reticle because the mental arithmetic is much simpler with decimal units, thereby requiring less mental calculation in the field. Using the range estimation formula with the units meters for range and millimeters for target size it is just a matter of moving decimals and do the division, without the need of multiplication with additional constants, thus producing fewer rounding errors.

${\displaystyle {\text{distance in meters}}={\frac {\text{target in millimeters}}{\text{angle in mils}}}}$

The same holds true for calculating target distance in kilometers using target size in meters.

${\displaystyle {\text{distance in kilometers}}={\frac {\text{target in meters}}{\text{angle in mils}}}}$

If using the imperial units yards for distance and inches for target size, one has to multiply by a factor of 100036 ≈ 27.78, since there are 36 inches in one yard.

${\displaystyle {\text{distance in yards}}={\frac {\text{target in inches}}{\text{angle in mils}}}\times 27.78}$

Also, in general the same unit can be used for subtension and range if multiplied with a factor of thousand, i.e.

${\displaystyle {\text{distance in meters}}={\frac {\text{target in meters}}{\text{angle in mils}}}\times 1,000}$

### Examples

Land Rovers are about 3 to 4 m long, "smaller tank" or APC/MICV at about 6 m (e.g. T-34 or BMP) and about 10 m for a "big tank." From the front a Land Rover is about 1.5 m, most tanks around 3 - 3.5 m. So a SWB Land Rover from the side are one finger wide at about 100 m. A modern tank would have to be at a bit over 300 m.

If for instance a target known to be 1.5 m wide (1500 mm) is measured to 2.8 mils in the reticle, the range can be estimated to:

${\displaystyle {\text{distance in meters}}={\frac {1500~{\text{mm}}}{2.8~{\text{mils}}}}=535.7~{\text{m}}}$

So if the above-mentioned 6 m long BMP (6000 mm) is viewed at 6 mils its distance is 1000 m, and if the angle of view is twice as large (12 mils) the distance is half as much, 500 m.

When used with some riflescopes of variable objective magnification and fixed reticle magnification (where the reticle is in the second focal plane), the formula can be modified to:

${\displaystyle {\text{distance in meters}}={\frac {\text{size in mm}}{\text{angle in mils}}}\times {\frac {\text{mag}}{10}}}$

Where mag is scope magnification. However, a user should verify this with their individual scope since some are not calibrated at 10×. As above target distance and target size can be given in any two units of length with a ratio of 1000:1.

## Definitions for maps and artillery

Map measure M/70 of the NATO member Denmark with the full circle divided into 6400 NATO mils
In the Swiss Army, 6400 "artillery per milles" ("Artilleriepromille") are used to indicate an absolute indication of direction by using the notation that 0 A ‰ (corresponding to 6400 A ‰) points to the north, instead of using NATO mils where direction is always relative to the target (0 or 6400 NATO mils is always towards target).

There are 2000π milliradians (≈ 6283.185 mil) in a circle; thus a milliradian is just under 16283 of a circle, or ≈ 3.438 minutes of arc. Each of the definitions of the angular mil are similar to that value but are easier to divide into many parts.

• 16283 The "real" trigonometric unit of angular measurement of a circle in use by telescopic sight manufacturers using stadiametric rangefinding in reticles.
• 16400 of a circle in NATO countries.
• 16000 of a circle in the former Soviet Union and Finland (Finland phasing out the standard in favour of the NATO standard).
• 16300 of a circle in Sweden. The Swedish term for this is streck, literally "line". Sweden (and Finland) have not been part of NATO nor the Warsaw Pact. Note however that Sweden has changed its map grid systems and angular measurement to those used by NATO, so the "streck" measurement is obsolete.

### Conversion table

Conversions between units
Milliradian NATO mil Warsaw Pact Mil Swedish streck Degrees Minute of arc
1 milliradian = 1 1.018592 0.954930 1.002677 0.057296 3.437747
1 NATO mil = 0.981719 1 0.9375 0.984375 0.05625 3.375
1 Warsaw Pact mil = 1.047167 1.066667 1 1.05 0.06 3.6
1 Swedish streck = 0.997302 1.015873 0.952381 1 0.057143 3.428572
1 degree = 17.452778 17.777778 16.666667 17.5 1 60
1 minute of arc = 0.290880 0.296297 0.277778 0.291667 0.016667 1

(Values in bold face are exact.)

• 1 trigonometric milliradian (mil) ≈ 3.43774677078493′
• 1 NATO mil = 3.375′ exactly
• 1 Warsaw Pact mil = 3.6′ exactly

### Use in artillery sights

Artillery uses angular measurement in gun laying, the azimuth between the gun and its target many kilometres away and the elevation angle of the barrel. This means that artillery uses mils to graduate indirect fire azimuth sights (called dial sights or panoramic telescopes), their associated instruments (directors or aiming circles), their elevation sights (clinometers or quadrants), together with their manual plotting devices, firing tables and fire control computers.

Artillery spotters typically use their calibrated binoculars to walk fire onto a target. Here they know the approximate range to the target and so can read off the angle (+ quick calculation) to give the left/right corrections in metres.

## References

1. ^ Renaud, Hugues (2002-05-31). Dictionnaire historique de la Suisse. Fonds, AV Laussane. Dapples: ... Charles-Marc (1837-1920), ingénieur, professeur à l'université de Lausanne, municipal à Lausanne, est l'inventeur de l'unité appelée "millième" pour mesurer les angles dans le tir d'artillerie. Une branche de la famille s'est fixée à Gênes à la fin du XVIIIe s.